Total curvature of graphs after Milnor and Euler

Robert Gulliver and Sumio Yamada

Vol. 256 (2012), No. 2, 317–357

DOI: 10.2140/pjm.2012.256.317

Abstract

We define a new notion of total curvature, called net total curvature, for finite graphs embedded in ℝⁿ, and investigate its properties. Two guiding principles are given by Milnor’s way of measuring using a local Crofton-type formula, and by considering the double cover of a given graph as an Eulerian circuit. The strength of combining these ideas in defining the curvature functional is that it allows us to interpret the singular/noneuclidean behavior at the vertices of the graph as a superposition of vertices of a 1-dimensional manifold, so that one can compute the total curvature for a wide range of graphs by contrasting local and global properties of the graph utilizing the integral geometric representation of the curvature. A collection of results on upper/lower bounds of the total curvature on isotopy/homeomorphism classes of embeddings is presented, which in turn demonstrates the effectiveness of net total curvature as a new functional measuring complexity of spatial graphs in differential-geometric terms.

Keywords

spatial graphs, total curvature, Milnor

Mathematical Subject Classification 2010

Primary: 05C99, 53A04, 57M25, 57N45

Milestones

Received: 30 April 2011

Revised: 8 July 2011

Accepted: 19 July 2011

Published: 30 May 2012